Solve Equations with Fractions: example 4.107

Translate and solve: The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Translate and solve. The quotient of $n$ and $\frac{2}{3}$ is $\frac{5}{12} .$

How did it go?

		The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$ .
Translate.		$\frac{m}{\frac{5}{6}} = \frac{3}{4}$
Multiply both sides by $\frac{5}{6}$ to isolate $m$ .		$\frac{5}{6} (\frac{m}{\frac{5}{6}}) = \frac{5}{6} (\frac{3}{4})$
Simplify.		$m = \frac{5 \cdot 3}{6 \cdot 4}$
Remove common factors and multiply.		$m = \frac{5}{8}$
Check:
Is the quotient of $\frac{5}{8}$ and $\frac{5}{6}$ equal to $\frac{3}{4}$ ?	$\frac{\frac{5}{8}}{\frac{5}{6}} \overset{?}{=} \frac{3}{4}$
Rewrite as division.	$\frac{5}{8} \div \frac{5}{6} \overset{?}{=} \frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.	$\frac{5}{8} \cdot \frac{6}{5} \overset{?}{=} \frac{3}{4}$
Simplify.	$\frac{3}{4} = \frac{3}{4} ✓$

		The quotient of $m$ and $\frac{5}{6}$ is $\frac{3}{4}$ .
Translate.		$\frac{m}{\frac{5}{6}} = \frac{3}{4}$
Multiply both sides by $\frac{5}{6}$ to isolate $m$ .		$\frac{5}{6} (\frac{m}{\frac{5}{6}}) = \frac{5}{6} (\frac{3}{4})$
Simplify.		$m = \frac{5 \cdot 3}{6 \cdot 4}$
Remove common factors and multiply.		$m = \frac{5}{8}$
Check:
Is the quotient of $\frac{5}{8}$ and $\frac{5}{6}$ equal to $\frac{3}{4}$ ?	$\frac{\frac{5}{8}}{\frac{5}{6}} \overset{?}{=} \frac{3}{4}$
Rewrite as division.	$\frac{5}{8} \div \frac{5}{6} \overset{?}{=} \frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.	$\frac{5}{8} \cdot \frac{6}{5} \overset{?}{=} \frac{3}{4}$
Simplify.	$\frac{3}{4} = \frac{3}{4} ✓$